A remark on fullness of some group measure space von Neumann algebras

Abstract

Abstract. Recently C. Houdayer and Y. Isono have proved among other things that every biexact group Γ has the property that for any non-singular strongly ergodic essentially free action Γ (X, µ) on a standard measure space, the group measure space von Neumann algebra Γ ⋉ L ∞ (X) is full. In this note, we prove the same property for a wider class of groups, notably including SL(3, Z).

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

“…gives a lattice isomorphism from P(n−2) onto L. By Theorem 11 of [47], all intermediate von Neumann subalgebras of N ⊂ M but N are full. Note that G/G S is identified with the partial flag manifold of signature t 1 , .…”

“…Hence, by Corollary 3.8, the amenable subfactor L ∞ (Fl 3 (R))⋊ Γ ⊂ L ∞ (P 2 R) ′⋊ Γ has no proper intermediate von Neumann algebras. It is worth mentioning that Ozawa [47] recently showed that L ∞ (P 2 R)⋊ Γ, hence L ∞ (P 2 R) ′⋊ Γ, is a full factor.…”

The approximation property and exactness of locally compact groups

“…Recent researches (see e.g., [1], [2], [21], [27], [32], [33]) show that non-discrete locally compact groups also provide rich sources of interesting operator algebras. They also reveal attractive and fruitful interactions between locally compact group theory and theory of operator algebras.…”

CAT(0) cube complexes and inner amenability

“…Similar connections between inner amenability and central sequences were later found by Choda [12] and Jones and Schmidt [21] in the context of ergodic theory. These connections to operator algebras and ergodic theory have continued to provide a rich context and motivation for the study of inner amenability; see, e.g., [38,23,25,24,11,26,27,37,19,33,15,20,3,22,28]. Perhaps because of this, inner amenability has been studied primarily by virtue of its relevance to these two fields (with a few exceptions, e.g., [1,2,36,18]).…”